Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 29K / a / Solution

A european call option with maturity $N$ and strike $K$ paysat time $N$. Let $q$ be the risk-neutral conditional probability of the factor $1+b$. The discounted stock must be a martingale, soand thereforeBoth probabilities are strictly positive. At every node these are the unique probabilities satisfying the martingale condition, so the fundamental theorem of asset pricing gives a unique no-arbitrage price.

If exactly $n$ of the $N$ moves use the factor $1+a$, thenand there are $\left(\genfrac{}{}{0ex}{}{N}{n}\right)$ such paths. Discounted risk-neutral expectation now giveswhere

Solved by gpt-5.6-sol high.